Trigonometric equations practice · Easy

The correct answer is B (8) because solving the trigonometric equation $\cos x \cos 2x \cos 3x = 1/4$ within the interval $[0, 2\pi]$ yields 8 distinct solutions.

Option A (5) is correct because solving the compound inequality $5x + 1 > -24$ and $5x - 1 < 24$ results in $-5 < x < 5$, which corresponds to the interval $(-5, 5)$.

Option B (4) is correct because there are four solutions for x in the interval (-$\pi$, $\pi$) that satisfy the given equation.

The number of solutions is 3 because solving the equation leads to $\sin x = \frac{1}{2}$ and $\sin x = -1$, giving three solutions in the interval $[0, 2\pi)$.

Option B (18) is correct because applying AM >= GM inequality, the minimum value of the expression is 18.

The correct answer is C ($\pi / 3$) because substituting $x = \sin \theta$ and simplifying the expression leads to this constant value.

Option D (4) is correct because there are four values of $\theta$ in the given interval that satisfy the equation.

Option C (Rational number) is correct because for the equation to have at least one solution, 'a' must be a rational number, ensuring that the arguments of sine and cosine are related in a way that allows both terms to reach their maximum values simultaneously.

Option B ($x = 2n\pi \, \pm \,{\cos ^{ - 1}}\,\left( {\frac{1}{3}} \right)$) is correct because it represents the general solution of the given trigonometric equation.

Option A ($n\\;\pi \\, + \\,{\left( { - 1} \right)^n}\frac{\pi }{4},\\,n \in Z$) is correct because it represents the general solution for $\theta$ derived from the given equation $\sec \theta \tan \theta = \sqrt 2$.

Option C, $\frac{\pi }{4}$, is correct because it is the only solution that satisfies the given equation $\sec \,x\,\cos \,5x + 1 = 0$ within the interval $0 < x < 2\pi$.

Option C ($\cos \left( {\theta - \frac{\pi }{4}} \right) = \frac{1}{{2\sqrt 2 }}$) is correct because it is derived by using trigonometric identities to simplify the given equation.

Option A (2) is correct because solving the inequality $\frac{2x-1}{3} \geq \frac{3x-2}{4} - \frac{2-x}{5}$ results in $x \leq 2$, which corresponds to the interval $(-\infty, 2]$.

Option D (4) is correct because there are 4 solutions for x in the given interval that satisfy the logarithmic inequality.

Option D (No solution) is correct because the maximum value of $3 \cos x + 4 \sin x$ is 5, which is less than 6.

Option A is correct because by using the sum-to-product trigonometric identities and solving the resulting equations, we find the general value of $\theta$.

The correct answer is Obtuse angled because the given conditions lead to angles A and B such that one of the angles in the triangle is greater than 90 degrees.

Option B ($\{\frac{\pi}{3}, \frac{5 \pi}{3}\}$) is correct because these are the only solutions to the equation $(2 \cos x-1)(3+2 \cos x)=0$ within the given interval $0 \leq x \leq 2 \pi$.

Option C ($\pi / 6$) is correct because by rewriting the equation using trigonometric identities and solving for theta, we find that theta = pi/6.

Option B ($2 \leq a \leq 6$) is correct because the equation has a solution only when the value of $\sin x$ derived from the quadratic equation lies between -1 and 1.

Option C ($\frac{{n \pi}}{{8}}$) is correct because solving the trigonometric equation leads to $\sin 8 \theta = 0$, thus $\theta = \frac{{n \pi}}{{8}}$.

Option C ($n\mathrm{\pi }\\; \pm \\; {\rm{\frac{{\pi }}{6}}}$) is correct because it represents the general solution for the given trigonometric equation.

Option A ($\left( {2n + 1} \right)\frac{\pi }{4}$) is correct because it is derived from simplifying the trigonometric equation and solving for x.

Option B (-1) is correct because the product of the roots of the quadratic equation in $\tan \frac{{\theta}}{{2}}$ is -1.

Option A ($\{ {n\\;\pi \\,\pm \\,\frac{\pi }{{3}}}\\;:n \in Z \}$) is correct because it is derived by simplifying the given trigonometric equation and solving for x.

Option B ($x = 2n\pi \, \pm \, {\cos ^{ - 1}}(\frac{1}{3})$) is correct because it represents the general solution to the trigonometric equation $3 \sin^2 x + 10 \cos x - 6 = 0$ after simplification and solving for x.

Option A ($\phi$) is correct because the expression simplifies to $\sin^2 a$, which is independent of $\phi$.

Option A (6) is correct because there are 6 possible ordered pairs (x, y) that satisfy the given equation within the specified intervals.

Option D ($\left( {\frac{{5\pi }}{4},\;\frac{{7\pi }}{4}} \right)$) is correct because the solution to the equation ${\cos ^2}\theta + \sin \theta + 1 = 0$ is $\theta = \frac{{3\pi }}{2}$, which lies in this interval.

Option C ($\cos \left( {x + y} \right)$) is correct because it is the simplified form of the given expression using the cosine addition formula.